Syllabus Map
- Study map: Syllabus Study Map
Core Idea
- A linear system is written as .
- Rank tells how many independent constraints/features exist.
Key Facts
- Unique solution: full rank square system.
- Infinite solutions: at least one free variable.
- No solution: inconsistent equations.
Practical Notes
Rank deficiency implies redundant information.
- Correlated features can reduce effective dimensionality.
REF/RREF provide structural diagnostics.
- Use pivots to identify dependencies and solvability.
Why This Matters for ML
- Least-squares fitting and many closed-form estimators reduce to solving linear systems.
- Rank reveals multicollinearity and redundant features, which affect model identifiability.
- Underdetermined systems motivate regularisation and minimum-norm solutions.
- Pivot structure from REF/RREF gives a practical diagnostic for solvability and feature dependence.